Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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10 Most Recent Blog Comments:

Brandiwyn🎶

a day ago
In response to "Blond Man's Bluff"

That's interesting about the two spellings of blonde. I always thought blonde referred to the person and blond referred to the hair. But I never bothered to look it up, So you've catered to my lazy today.

3 Like

Kåre เลียม Enga

Nov 9, 2024 at 1:02pm
In response to "It Was a Dark and Starry Night"

Ah... but why did he see those swirls when many of us don't? A totally different discussion... out of thin air... which is a frozen fog today when I gaze out the window.

Humans have great skills of observation; some of us extraordinary skills. Is The Scream depictive or evocative? 😱

Turkey DrumStik

Nov 9, 2024 at 11:04am
In response to "It Was a Dark and Starry Night"

Frankly, I'm more surprised to see reasonably good reporting from CNN. I'd given up on their ability to rein in their tendency toward loaded speech.

2 Like

JayNaNoOhNo

Nov 8, 2024 at 12:00pm
In response to "Bungalow Bash"

For the love of God, what did I just read? That is by far the worst use case of a bungalow I've ever seen, and a fundamental misunderstanding of the value of front porches—and the size of bungalows.

Was this his point? Nope, I know what his point was. For a guy so worried about presentation, he certainly undercut his own arguement.

1 Like

Turkey DrumStik

Nov 8, 2024 at 9:47am
In response to "Bungalow Bash"

"People don't change their minds on the internet. They usually do that in books, battlefields, or not at all.

Yeah, that's a little bleak."

Bleak and also inaccurate.

1 Like

Kåre เลียม Enga

Nov 7, 2024 at 10:19pm
In response to "Past and Future Intents"

I must live in the present. In the past I daydreamed a lot... probably to escape the bruises of reality; however, I need to be more aware crossing streets or even how I get out of bed (no rapid movements without thinking of consequences).

I can envision a future more peaceful than the present because this time-and-place isn't the only "present" that's ever existed, and I do try to see the larger view (emphasis on 'try').

1 Like

Lilli 🧿 ☕

Nov 7, 2024 at 10:58am
In response to "Past and Future Intents"

If the aforementioned exercises don't help; there's still liquor.

3 Like

Turkey DrumStik

Nov 6, 2024 at 10:22am
In response to "Lo!"

Not gonna lie I need the holiday season chatter.

Aiva Raine

Nov 5, 2024 at 3:03pm
In response to "It's Hard Not To Scare"

Shaking head. Aside from the inevitable Simpson's spider goat song playing in my head which I will now share with you below, whenever I come across a story that has all these scientists doing ridiculous experiments for no logical reason, I must go- ah- scientists- CANCER? You do remember that problem right? Can we maybe cure that before we make cockroaches magnetic and elephants see pink elephants?

spider pig clip- and if you all now are singing the spider goat song- you're welcome! grin
https://www.youtube.com/watch?v=BARjPuUN36Y

1 Like

Aiva Raine

Nov 5, 2024 at 2:55pm
In response to "Anomalies"

One of your more humorous entries- I especially enjoyed- "Lofoten Vortex can be the name of my Bjork cover band."

Thanks for the laughs. Much appreciated.

2 Like

November 20, 2022 at 12:01am

#1040853

Revisited: "Rarity"

One of the reasons I've been randomly digging into the archives on occasion is that things change, especially me.

"Rarity"

is from mid-2009, and it made me wonder just how much some things actually change. The entry was done in what would eventually become my standard format, commenting on an article I found on the internet. Being from 2009, though, the actual article is gone, but hopefully there's enough in my entry to get the general idea: some employers were allegedly having trouble filling positions.

Like I said, made me wonder just how much some things actually change.

In 2009, the US was in the middle of the Great Recession, marked in part by high unemployment rates. It peaked then at about 10%, and that number would steadily decline over the next several years until it shot up again in early 2020 (gosh I wonder what happened then)—but we couldn't have known that at the time. The current value is around 3.5%, close to what it was in the Before Time.

With an unemployment rate that low, I can understand employers not being able to find workers. At 10%, though? Well, I guess the original article was about professionals and trade workers, not unskilled labor.

What stood out for me in my original entry above was this:

Businesses try to create a glut of workers so that they can have more control over the workers. Supply and demand, folks. If there are 30 engineers competing for one position, they'll go with the competent one who can live on the lowest salary. Just business. If, however, there are 30 possible positions for each engineer, the engineer's in the catbird's seat.

I'm not sure why I phrased it that way now, implying that businesses are creating workers somehow. I mean, I guess they do to some extent when they go to the press with "we have a shortage of [nurses|engineers|welders|whatever] so kids should go learn these trades!" But I can't see that making a huge difference.

What I probably should have said was that businesses like it when there's an abundance of workers. And I still think that way. Oversupply of workers leads to management deciding they can make more demands of them, while a scarcity of potential employees means someone looking for a job can be the one doing the demanding.

Embarrassing moment for me in the entry:

Let's play SAT test (don't worry; no math in this one).

Oh, I'll need to pull cash from the ATM machine and use my GPS system to get to the testing facility. Oy. Sorry about the redundancy. I make mistakes from time to time. I know, I know; it was hard to believe when I wrote yesterday that I don't always get everything right. But this is proof.

In any case, i was struck by the similarities of the 2009 article (at least what excerpts survived in my entry) and a lot of the rhetoric businesses are spouting today. I tried to find a similar article from this year or at least last year, to see what "the hardest jobs to fill" are right now, but my search came up with nothing relevant. I did find this

, but there's no date on the page (apart from a copyright that probably updates every January 1), and the salaries listed appear to be from 20 years ago. So I have no idea what the hardest jobs to fill right now are.

I suspect "blogger" isn't on the list, though.

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